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Among the many proposals to approach the concept of time in quantum theory,the Page-Wootters mechanism has attracted much attention in the last few years.Originally, such a mechanism explored a stationary bipartite non-interactingglobal system, i.e., a system of interest together with an ancillary clock, todetermine how the evolution in time can emerge as an equation of motion for aquantum particle conditioned to the measurement of the state of the clock.After the seminal proposal, many variations of it were considered, anddifferent aspects of the mechanism were elucidated. Our contribution to theseinvestigations is that we analyze such a timeless approach to quantum theorybut deriving an equation of motion for a mixed state system that evolvesaccording to its gravitationally induced interaction with a non-ideal quantumclock. The interaction considered is known to describe the gravitationaldecoherence mechanism, and the clock model is the recently proposed quasi-idealclock, i.e., one constructed to approximate the time-energy canonicalcommutation relation. As a result of our considerations, we obtained anequation of motion that is non-linear in nature, dependent on the system'sinitial conditions.

Non-linear equation of motion for Page-Wootters mechanism with interaction and quasi-ideal clocks